A function defined by , where , is called the logarithm of to the base . The natural logarithmic function is written as or .
We shall prove the formula for the derivative of the natural logarithm function using definition or the first principle method.
Let us suppose that the function is of the form
First we take the increment or small change in the function:
Putting the value of function in the above equation, we get
Dividing both sides by , we get
Multiplying and dividing the right hand side by , we have
Taking the limit of both sides as , we have
Consider , as then , we get
Using the relation from the limit , we have
Example: Find the derivative of
We have the given function as
Differentiating with respect to variable , we get
Using the rule, , we get